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Homotopical patch theory

2014
*
Proceedings of the 19th ACM SIGPLAN international conference on Functional programming - ICFP '14
*

A

doi:10.1145/2628136.2628158
dblp:conf/icfp/AngiuliMLH14
fatcat:oqd3onn2jvh3nnbipg4hfi7dne
*patch**theory*is presented by a higher inductive type. Models of a*patch**theory*are functions from that type, which, because function are functors, automatically preserve the structure of*patches*. ... We reformulate*patch**theory*using the tools of homotopy type*theory*, and clearly separate formal*theories*of*patches*from their interpretation in terms of basic revision control mechanisms. ... Building on this work, we develop*patch**theory*in the context of homotopy type*theory*, using paths to model aspects of*patch**theory*. ...##
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Planar Graphs on Nonplanar Surfaces

1996
*
Journal of combinatorial theory. Series B (Print)
*

Suppose now that at each vertex of C,

doi:10.1006/jctb.1996.0058
fatcat:lvlzteyqbfcbxloihhxzisisgi
*patch*faces are only on one side of C. Then there is a cycle C*homotopic*to C that contains only vertices in the interiors of*patches*. ... Since # cannot escape out of D, it is*homotopic*to # 1 and it must intersect # 1 (or # 2 ) twice. Let x$ be the other*patch*vertex used by #. ...##
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Page 305 of Mathematical Reviews Vol. 57, Issue 1
[page]

1979
*
Mathematical Reviews
*

He then formulates ‘‘a simple rule for identifying

*theories*that admit physically significant*patching*ambiguities”. ... Preprint, Harvard Univ., Cam- bridge, Mass., 1975] has given rules for*patching*together two topologically stable field configurations in non-Abelian gauge*theories*(that is, constructing models of a two-particle ...##
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More (thoughts on) Gribov copies

1992
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Nuclear Physics B
*

In ref. [6] we proposed formulating the hamiltonian

doi:10.1016/0550-3213(92)90386-p
fatcat:wfdt6tdp3fgltgnziihzqsavuy
*theory*on coordinate*patches*with*homotopically*non-trivial gauge transformations as transition functions. ... We can shrink these*patches*almost to A (and their associated gauge copies, with the*homotopically*non-trivial gauge transformations that relate the inequivalent classical vacua). ...##
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Tōhoku

2015
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Inference: International Review of Science
*

W e now understand the

doi:10.37282/991819.15.13
fatcat:3pmvwf64pzc5vddgj5t7zrstfu
*theory*of stacks and non-abelian cohomology*theory*in a much more tractable,*homotopical*way. Stack*theory*is, I suggest, the modern geometric*theory*of symmetries. ... We end with a purely*homotopical*approach to the construction of the derived category of abelian sheaves, one that is consistent with the*homotopical**theory*of simplicial sheaves and presheaves in the ...##
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Homotopy Transition Cocycles
[article]

2006
*
arXiv
*
pre-print

The full

arXiv:math/0609220v2
fatcat:p7zdry3nenaw5h3uhvhcsxsboq
*theory*was worked out by the first author in his 1965 Notre Dame thesis wirth:diss. Here we present it using language that has been developed in the interim. ... trivial fibrations, one can define transition functions g : U∩ U→ H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that g g is*homotopic*... In Section 5, Wirth's concept of a "fibration*theory*" is axiomatized. Here too the*patching*theorem is crucial. ...##
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An Approximation Theorem for Maps Between Tiling Spaces
[article]

2018
*
arXiv
*
pre-print

If two local maps are

arXiv:0906.4741v2
fatcat:fvuoiduydrfulgpdaz7t5hdurm
*homotopic*, then the homotopy can be chosen so that every interpolating map is also local. ... Introduction Many aspects of tiling*theory*, such as pattern-equivariant cohomology [2, 3] , are built around local data. ... central*patches*of radius 1. ...##
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Special issue dedicated to ICFP 2014: Editorial

2016
*
Journal of functional programming
*

demonstrate both the quality and the breadth of the conference, with a strong emphasis on types and their applications, and ranging from compilation methods through contract verification to homotopy type

doi:10.1017/s0956796816000228
fatcat:raf3t4hb3jahthr3ngcb5oo6zq
*theory*... In*Homotopical**Patch**Theory*, Angiuli, Morehouse, Licata, and Harper consider a programming application of higher inductive types, a new class of datatypes that arises in homotopy type*theory*. ... The paper considers a sequence of*patch**theories*, including two that did not appear in the original conference version of the paper, culminating in a*patch**theory*of text files. ...##
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Topological Membrane Solitons and Loop Orders of Membrane Scatterings in M-theory
[article]

1996
*
arXiv
*
pre-print

*Patching*of these solitons and their topological charges are also defined and discussed. (2) Loop order of membrane scatterings is the basis for a perturbative M-

*theory*. ... Two topological issues on membranes in M-

*theory*are studied: (1) Soliton is an important subject in M-

*theory*. ... The additivity property of charges under

*patchings*still holds. As a comparison, the

*patching*of membrane solitons defined here generalizes the

*patching*that occurs in gauge

*theory*. ...

##
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Page 548 of The American Mathematical Monthly Vol. 58, Issue 8
[page]

1951
*
The American Mathematical Monthly
*

Whereas a cycle is homologous to zero if it bounds, it is

*homotopic*to zero if it can be contracted into a point by continu- ous deformation. ... So much, however, seems clear that one had better start, not with a division into cells, but with a covering by*patches*which are allowed to overlap. ...##
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Global Color Is Not Always Defined

1983
*
Physical Review Letters
*

Far away from a fundamental magnetic monopole of the SU(5) grand unified

doi:10.1103/physrevlett.50.943
fatcat:aohk5swrkzhkxm5zr7h7bw7nwi
*theory*, the full*theory*reduces to one whose gauge group is unbroken SU(3) ⊗ U(1). ... Disciplines Physical Sciences and Mathematics | Physics Comments ABSTRACT Far away from a fundamental magnetic monopole of the SU(5) grand unified*theory*, the full*theory*reduces to one whose gauge group ... To proceed beyond this heuristic level it is convenient to 4 describe the unbroken*theory*in Wu and Yang's two-*patch*formalism.The*theory*is defined on the space 1R x OR 3 -B }, where B is a ball occupied ...##
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Page 8930 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

Some important flows such as vortex

*patches*belong to this class.” J. ... Near the absolute instability threshold, spatial growth rates are larger than those predicted by temporal stability*theory*. ...##
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Page 4428 of Mathematical Reviews Vol. , Issue 2003f
[page]

2003
*
Mathematical Reviews
*

*Theory*Appl. 114 (2002), no. 3, 609-637. ... This attempt opens a route which, on further examination, might eventually reach unexplored

*patches*in the field. ...

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Global signatures of gauge invariance: Vortices and monopoles

1976
*
Physical Review D, Particles and fields
*

By way of homotopy

doi:10.1103/physrevd.14.1006
fatcat:7gheplrs6ngktfdct4qzics4za
*theory*, a simple analysis is presented for the global groups U (1)) O(3), SU(2) then SU(N)/ZN and SU(N). ... A comprehensive topological classification of vortices and their endpoint Dirac monopoles is formulated in gauge*theories*with an arbitrary compact Lie group. ... One of us (HCT) wishes to thank his colleagues at the SLAC*Theory*Group for many discussions. ...##
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Desingularizing homology manifolds

2007
*
Geometry and Topology
*

In earlier work, [4] , we showed that nonresolvable homology manifolds exist and can be classified, up to s-cobordism, by a variant of surgery

doi:10.2140/gt.2007.11.1289
fatcat:g7zbf3btgzcu5jguljwzh7yrra
*theory*. ... We may assume that f restricts to U V 1 maps on the*patches*of X . ...
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